Each application can have different requirements for the wire. For example, wire used in die-attach applications needs tight dimensional tolerances to insure an exact, repeatable amount of solder is deposited each time. Reduced oxides are also critical to eliminate any "splattering" of the molten solder during the deposition process.
Wire can also be used for non-soldering applications. For example, indium (and indium alloys) wire are often used as a sealing material (particularly in cryogenic sealing applications) - more here) and as a thermal interface / management material.
Decades ago, 0.030" (0.76mm) diameter was the standard size, but today we are able to produce diameters as small as 0.001" (0.025mm) in tin silver (Sn Ag), tin silver copper (SAC) and gold tin (Au Sn) alloys. Considering that a human hair is about 4X that size, that is a very small diameter! Pure indium wire is limited to 0.010" (0.254mm), but alloys containing indium can be produced smaller than that.
The wide variety of diameters available in Au Sn make this alloy ideal for the complex applications in medical, aerospace, and other high reliability applications. However, the Sn Ag and the Sn Ag Cu are used across a variety of standard applications that require lead-free materials. Sn Ag is particularly good in soldering to Nitinol.
At first look, wire seems like a pretty simple product. But specifying the right alloy, diameter, tolerances, and packaging can make all the difference. It can help you achieve a repeatable process that gives you high yields, strong solder joints, and enhanced profitability.